Investigating the dependence of collective dynamics on n/p asymmetry for light nuclei
R.T. deSouza, Varinderjit Singh, S. Hudan, Z. Lin, C.J. Horowitz
aa r X i v : . [ nu c l - e x ] M a y Investigating the dependence of collective dynamics on n/p asymmetry for light nuclei
R. T. deSouza, ∗ Varinderjit Singh, and S. Hudan
Department of Chemistry and Center for Exploration of Energy and Matter, Indiana University2401 Milo B. Sampson Lane, Bloomington, Indiana 47408, USA
Z. Lin
Department of Physics and Center for Exploration of Energy and Matter, Indiana University2401 Milo B. Sampson Lane, Bloomington, Indiana 47408 USA andDepartment of Physics, Arizona State University,450 E. Tyler Mall, Tempe, AZ 85287-1504 USA
C. J. Horowitz
Department of Physics and Center for Exploration of Energy and Matter, Indiana University2401 Milo B. Sampson Lane, Bloomington, Indiana 47408, USA (Dated: May 8, 2020)The dynamics present in the fusion of neutron-rich nuclei is explored through the comparison ofexperimental cross-sections at above-barrier energies with measurements of the interaction cross-section at relativistic energies. The increase of fusion dynamics with increasing neutron excess isclearly demonstrated. Experimental cross-sections are compared with the predictions of a Sao Paulomodel using relativistic mean field density distributions and the impact of different interactions isexplored.
PACS numbers: 21.60.Jz, 26.60.Gj, 25.60.Pj, 25.70.Jj
Nuclei are extremely interesting quantal systems,which despite a limited number of constituent particles,manifest collective dynamics. This collective dynamics isobserved in many forms including the giant dipole res-onance [1, 2], shape coexistence [3], and fission [4, 5].Although typically associated with the structure and re-actions of mid-mass and heavy nuclei, collectivity for verylight nuclei has recently been reported [6]. Nuclear fis-sion and nuclear fusion provide examples in which col-lective degrees of freedom undergo substantial change asthe reaction proceeds. Of particular interest is the roleof collectivity for neutron-rich nuclei as for these nucleithe dependence of the dynamics on the asymmetry be-tween the neutron and proton densities can be probed.Fusion reactions provide a powerful means to assess theresponse of neutron-rich nuclei to perturbation. As fu-sion involves the interplay of the repulsive Coulomb andattractive nuclear potentials, by examining fusion for anisotopic chain one probes the neutron density distribu-tion and how that density distribution evolves as thetwo nuclei approach and overlap [7, 8]. In the followingmanuscript we propose a novel perspective for investigat-ing the role of collective dynamics in fusion. Moreover,using this new perspective we elucidate the dependenceof the fusion dynamics on n/p asymmetry for the firsttime including the indication that for light nuclei fusiondynamics is enhanced with increasing neutron number.Measurement of the interaction cross-section, σ Int. inhigh energy collisions is an effective means to invesitgatethe spatial extent of the matter distribution [9]. The in-teraction cross-section in these measurements is simplydefined as the total nuclear reaction cross-section result- ing in a change of either the atomic number (Z) or massnumber (A) of the projectile. Systematic comparison ofthese cross-sections for lithium isotopes revealed the halonature of Li [10, 11]. Presented in Fig. 1 are the in-teraction cross-sections of carbon isotopes with a carbontarget. Measurements for A ≥
12 were made at E/A ∼ σ Int. with neutron excess, (N-Z). At the highincident energy that these experiments were conductedat, one expects the sudden approximation to be valid.Hence, the measured interaction cross-section, σ Int. pro-vides a direct measure of the extent of the matter dis-tribution. Through comparison with a Glauber model,the rms matter radii of these nuclides has been extracted[14].Closer examination of Fig. 1 provides an indication ofthe impact of shell structure on σ Int. . The dependenceof σ Int. on neutron excess for 12 ≤ A ≤
14 is weak as is thedependence for 16 ≤ A ≤
18. Between C and C one ob-serves a jump in σ Int. from a value of ∼
850 mb to ∼ withN=8 and the population of the 1d shell. This observa-tion is significant as it indicates that the shell structureof the neutron-rich isotopes is accessible through mea-surement of σ Int. for an isotopic chain.Juxtaposed with the measured interaction cross-sections in Fig. 1 are the results of calculations of the
N-Z − ( m b ) I n t. σ C +
A Int. σ (LBL) Int. σ RMF-SP(FSUGOLD)RMF-SP(NL3) ( m b )
17 14 〉 F u s i on σ 〈 FIG. 1. Comparison of the interaction cross-section σ Int. forvarious carbon isotopes with the prediction of the averagefusion cross-section at above-barrier energies using a RMF-SP model. See text for details. C C C C C C Radius0 1 2 3 4 5 60.020.040.060.080.10.12 C C Radius N eu t r on D en s i t y FIG. 2. Neutron density distributions predicted by the RMFmodel with the FSUGOLD interaction for carbon isotopes. fusion cross-section at energies just above the fusion bar-rier. These calculations were performed at energies ofE CM = 1-2 MeV/A and utilize the Sao Paulo model forcalculating the fusion cross-section [15]. In this model thedensity distributions of the colliding nuclei are assumedto be frozen during the fusion process thus the calcu-lated cross-sections reflect the size of the colliding nuclei.Moreover, use of a common target nucleus allows one toassess the change in the size of the projectile nucleus withincreasing neutron excess. The density distributions usedin the Sao Paulo calculations were determined using arelativistic mean field (RMF) model [16, 17]. In order toinvestigate the sensitivity of the calculated cross-sectionsto the interaction used in the RMF calculations, the RMFcalculations were performed using two sets of interactionsFSUGOLD and NL3 [18]. In contrast to the widely usedNL3 interaction, the FSUGOLD corresponds to a softerinteraction [19]. To facilitate the comparison of the fusioncross-sections predicted by the RMF-SP model with σ Int. we have calculated the average fusion cross-section overthe interval 14 MeV ≤ E CM ≤
17 MeV and designate thisquantity <σ fusion > . Although the calculations withthe FSUGOLD interaction manifest a consistently largerfusion cross-section, nonetheless both RMF-SP calcula-tions exhibit a similar dependence on neutron number.For N < Z the value of <σ fusion > is approximately con-stant while for N > Z it increases approximately linearlywith (N-Z). The slope of the predicted cross-sections forthe two interactions shown is to first order the same indi-cating that while the absolute size of the nucleus dependson the interaction used in the RMF model, the increasein size with increasing N is relatively insensitive to theinteraction utilized. Moreover for N > Z, the slope of thepredicted above-barrier fusion cross-section is very sim-ilar to that for σ Int. . This similarity of the two slopesarises from the fact that the σ Int. measures the size ofthe nucleus and the RMF-SP with the frozen density dis-tributions is intrinsically related to the same quantity.
As such, the quantity σ Int. provides a key reference fromwhich to examine fusion dynamics.
In Fig. 2 the density distributions for neutrons pre-dicted by the RMF model for the various carbon iso-topes are displayed. The density distributions for N =Zare compared with that of C for reference. As expected,the tail of the neutron density distribution extends fur-ther for the more neutron-rich the isotope. The valueof calculating these neutron density distributions for anisotopic chain lies in the ability to examine the system-atic dependence on neutron number. The evolution ofthe nuclear size on neutron number may have differentsensitivity to the model uncertainties as compared to theabsolute size. Presented in Fig. 3 are the density dis-tributions for protons predicted by the RMF model forthe different isotopes of carbon. While the distributionsare all quite close as might be expected, as the isotopebecomes more neutron-rich the tail of the proton distri- C C C C C C Radius0 1 2 3 4 5 60.020.040.060.080.10.12 C C Radius P r o t on D en s i t y FIG. 3. Proton density distributions predicted by the RMFmodel with the FSUGOLD interaction for carbon isotopes. bution extends slightly further out. This change in theproton distribution is due to the attractive nuclear forceof the valence neutrons. The dependence of the chargeradii on neutron number for the carbon isotopic chainhas recently been determined through measurement ofthe charge changing cross-section [14].To investigate the evolution of fusion dynamics withincreasing neutron number we examine the fusion cross-section for A C+ C at near barrier energies. Using anovel active target approach the fusion excitation func-tions for these reactions was measured by the ANL group[22]. This active target approach is particularly wellsuited to studying reactions with low intensity beams andallowed measurement of the fusion excitation functionwith beam intensities as low as 500 ions/s. Depicted inFig. 4 (both columns) are the fusion cross-section datafor − C. The data for , C have been taken from[23]. Using the same approach though at higher beam in-tensity, , C cross-sections were also measured [20, 21]and are shown in Fig. 4. The fusion excitation functionsobserved for , C + C are in good agreement withthose published in the literature [24, 25]. The measuredexcitation functions manifest the expected dependence C C + C C + C C + C C + C C + C C + C C + C C + σ 〉
Fusion σ〈 RMF-SP(NL3)RMF-SP(FSUGOLD) (MeV) c.m. E ( m b ) σ FIG. 4. Fusion excitation functions for − C + c. Ex-perimental data are compared with the results of a RMF-SPmodel using FSUGOLD (left column) and NL3 (right col-umn). The fusion excitation function for C (not shown) iscomparable to that of C [20, 21]. indicative of a barrier driven process. The experimentaldata are compared with the results of the RMF-SP modelwith the FSUGOLD and NL3 interactions depicted in theleft and right columns respectively. While in the case of C the models overpredict the experimental results, inthe remainder of the cases the agreement is reasonable.Close comparison of the left and right columns indicatesthat the calculations with FSUGOLD consistently pre-dict larger cross-sections than those with NL3, consistentwith the observation in Fig. 1. It is noteworthy thoughthat this increase in the cross-section for FSUGOLD ascompared to NL3 is typical of the entire above-barrierregime. In Fig. 4 the blue bar indicates the value of theaverage cross-section as well as the energy interval overwhich the average was calculated. For − C the aver-age cross-section is clearly representative of the above-barrier cross-section. In the case of C, however there issignificant variation in the measured cross-sections andthe average cross-section calculated is more sensitive tothe choice of energy interval.In Fig. 5 the average above-barrier fusion cross-sectionsfor the carbon isotopic chain are compared with σ Int. .One observes that for C the fusion cross-section and σ Int. are essentially the same. For N > Z however, thefusion cross-section depends more strongly on neutronexcess than σ Int. does. Since the dependence of σ Int. onincreasing neutron number indicates the inherent growthin the size of the neutron density distribution with in-creasing neutron number, the cross-section in the case offusion, above that of σ Int. reflects the impact of dynamicsin the fusion process. The quantity ( <σ fusion > - σ Int. )can be viewed as a measure of the fusion dynamics. More-over, the increase in this quantity, dictated by the largerslope for fusion as compared to σ Int. , indicates that thisdynamics evolves with increasing neutron excess.It is also instructive to compare the behavior of the fu-sion data with the results of the RMF-SP model. The ex-perimental data indicates a stronger dependence on neu-tron excess than the RMF-SP model independent of theinteraction chosen. This comparison thus also indicatesan increased role for dynamics with increasing neutronexcess. Thus, the increased role of fusion dynamics withneutron excess is realized in two independent ways. Jux-taposition of the <σ fusion > with σ Int. , namely a datato data comparison, indicates enhanced fusion dynamicswith increasing neutron excess. This result is supportedby the comparison of <σ fusion > with the cross-sectionspredicted by the RMF-SP model.Although the span of neutron excess for the fusiondata presented is presently limited, the new generationof radioactive beam facilities allows one to extend thesemeasurements to even more neutron-rich carbon isotopes.Measurement of fusion with beams of , C and possi-bly , C at FRIB [26] is envisoned. Similar measure-ment for the oxygen isotopic chain extending nearly tothe neutron-drip line is also possible.Examination of the fusion cross-section at above-barrier energies for an isotopic chain is a powerful tool.Comparison of fusion cross-sections just above the barrierwith the interaction cross-section, σ Int. , at high energieswhere the sudden approximation is valid allows extrac-tion of not just the fusion dynamics but the dependenceof the dynamics on neutron excess. Investigating this dy-namics for the most neutron-rich nuclei accessible couldprovide valuable insight into the dynamics of extremelyasymmetric nuclear matter.This work was supported by the U.S. Department ofEnergy under Grant No. DE-FG02-88ER-40404 (In-diana University). CJH is supported in part by U.S.DOE grants DE-FG02-87ER40365 and de-sc0018083. ZLgratefully acknowledges support from National ScienceFoundation under PHY-1613708 (Arizona State Univer-sity).
N-Z − ( m b ) I n t. σ C C +
A Fusion σ Int. σ RMF-SP(FSUGOLD)RMF-SP(NL3) ( m b )
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